On 2/14/2013 12:40 PM, fom wrote:> On 2/14/2013 9:32 AM, Shmuel (Seymour J.) Metz wrote:>> In <qImdnYCz5tRmvITMnZ2dnUVZ_oWdnZ2d@giganews.com>, on 02/11/2013>> at 10:53 AM, fom <fomJUNK@nyms.net> said:>>>> You really need to step back, separate out the philosophy from the>> mathematics and define any terms that you aren't uisng in accordance>> with standard practice.>>

What follows are the difference setsfor each letter of the alphabet. Thefirst entry of each list is the "givenletter". This entry is followed by asign of equality. The "given letter"is not a member of its difference set.

The construction is a {96,20,4} design.That is, 96 elements into blocks of 20that contain quadruples uniquely.

As this part of the construction had beenmotivated by the axiom of regularity inset theory, it is intuitively reasonableto think of a letter as "a collectionof letters" although I suspect many willfind that objectionable.

These difference sets have numerousproperties. The free orthomodular latticeon 2 generators contains 6 Boolean blocksin the configuration of the 16-elementfree Boolean lattice on six elements.No element of a difference set is in theBoolean block of its "given letter".

One prior formation decision was achoice of a single column from amongthe six columns

TTTFFFTFFTTFFFTFTTFTFTFT

to correspond with a fixed representationof LEQ as a switching function. Becausethe letters have six distinct first coordinatesand sixteen distinct second coordinates,for any particular second coordinate of anyparticular letter, there are six correspondingletters. Each difference set partitionsone of these six-fold multiplicities intoa 5-set and a singleton. The singleton isin the same Boolean block as the "givenletter" for the difference set.

Because of the semantical relationshipsoriginating with truth-table semanticsthat motivated this 5-set can be takenas having the form of ortholattice L12depicted around equation (29) of

where the distinguished element of thelattice, namely c, can be taken to beXOR. The remaining four labels in thelattice can be arbitrarily assigned toFIX, LET, FLIP, and DENY. These arethe four switching functions invariantunder DeMorgan conjugation.

As for the singleton in the same blockas the "given letter" for the differenceset, it may be thought of as an "individuatingmark" in the sense of Leibnizian logic.Its binding to the "given letter" isrepresented by a Steiner Quadruple Systemon 14 symbols constructed from the other14 letters of the Boolean block in whichthe "given letter" and its "individuatingmark" are located.

An SQS(14) has 91 4-element blocks.

When the 5-set mentioned above is removedfrom the 96-elements of the free orthomodularlattice on two generators, there remainsa 91-element set. There are 4 projectiveplanes on 91 symbols. Although the constructionhas not yet been done, it is expected thatthe block designs can be formulated to expresssome uniformity in relation to one anotherand in relation to the SQS(14) formed relativeto the absence of the "given letter" and its"individuating mark".

In addition, the "given letter" and its"individuating mark" can be bound in aglobal sense relative to an

S_5{2,5,89}

design. This design locates any pairof letters into blocks of 5. The "S_5"prefix indicates that this relationshipof pairs to blocks will, itself, have amultiplicity of 5. Thus, every pair willoccur in 5 distinct blocks of 5.

Intuitively, these constructions areto be thought of as isolating the5-set mentioned above from the "givenletter" and its "individuating mark".Without the computational analysis ofhow related group actions affect theseconstructions, it is difficult to decideon how to describe the situation. Itis as if the 5-element set is the "object"and the "given label" with its "individuatingmark" comprise the presentation of"quantum duality".